CN112926384B - Automatic modal identification method based on power spectrum transfer ratio and support vector machine - Google Patents

Automatic modal identification method based on power spectrum transfer ratio and support vector machine Download PDF

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CN112926384B
CN112926384B CN202110055708.7A CN202110055708A CN112926384B CN 112926384 B CN112926384 B CN 112926384B CN 202110055708 A CN202110055708 A CN 202110055708A CN 112926384 B CN112926384 B CN 112926384B
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陈志为
刘奎铭
阮旭芝
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Abstract

The invention provides a modal automatic identification method based on a power spectrum transfer ratio and a support vector machine, which comprises the following steps: establishing a transfer ratio function and a transfer ratio matrix; calculating the gradient of a peak curve of the PSDT rational function; automatically identifying system poles based on a support vector machine; solving modal parameters to construct a stable graph; calculating a modal similarity coefficient; and automatically identifying a stable axis based on the support vector machine. The method provided by the invention can automatically operate to identify the modal parameters without manual intervention, can effectively eliminate false modes, can accurately identify the dense mode by the automatic identification method, and has strong robustness to resist noise interference, thereby having wide application prospect in real-time SHM.

Description

Automatic modal identification method based on power spectrum transfer ratio and support vector machine
Technical Field
The invention relates to the field of engineering structure health monitoring, in particular to a mode automatic identification method based on a power spectrum transfer ratio and a support vector machine.
Background
For long-term, on-line, real-time monitoring, Structural Health Monitoring Systems (SHMS) have been installed in many large buildings, such as large-span bridges and high-rise buildings. As an intrinsic property reflecting the overall characteristics of the structure, modal parameters can be efficiently identified from the vibration data collected by the SHMS and used to develop damage indicators for state assessment. The Operating Mode Analysis (OMA) technique is a correct method to identify modal parameters (e.g., natural frequency, damping ratio, and mode shape) of a structure based on the system response of the environmental stimulus. In early studies, experimental modal analysis techniques for input-output system identification were compared to OMA techniques. In general, experimental modal analysis methods are more reliable, but the OMA method has some advantages, such as applicability under operating conditions and considering the impact of boundary conditions on the results, and facilitates model updating.
The OMA method can be classified into a time domain and a frequency domain method. Frequency domain methods are widely used because of their high efficiency. The transfer ratio based OMA method (TOMA), first proposed by Devriendt and Guillaume, opens up a new way to solve the white noise assumption by directly determining the modal parameters using the valid information in the residual. It is less sensitive to the nature of the system input because it can effectively avoid measuring input data and modeling unknown inputs, and thus Sun et al further investigated two TOMA schemes. Following the work of Devrriendt et al, Yan and Ren propose the concept of power spectral density transfer ratio (PSDT) to identify the operating modal parameters under single input conditions and generalize conventional PSDT to the enhanced PSDT method by incorporating a least squares complex frequency domain (LSCF) estimator. Araujo proposes an OMA method (PSDTM-SVD) based on Singular Value Decomposition (SVD) of PSDT matrices with different references. Ara jo and Laier also extend the scalar PSDT concept to multi-reference point PSDT by correlating multiple transfer outputs. Li et al compared the dynamics of various transfer ratios and discussed the relationship between the single reference transfer ratio and the PSDT. Yan et al discussed previous studies that highlighted the future trends in current studies that remain inadequate and address transfer function-based system identification. Currently, the existing OMA methods using PSDT still require manual intervention, depending on the judgment and experience of the engineer. Thus, even though real-time vibration data is available for these structures equipped with SHMS, achieving both automation and accurate modal identification remains a significant challenge.
As one of the most interesting research topics in structural health monitoring, automatic modality identification has been widely studied and has made significant progress over the past few years, particularly in how to reduce manual intervention and eliminate spurious modalities. The automatic modal identification method can be mainly classified into two categories. In the first category, the significance index and its threshold are presented and discussed to eliminate spurious modes. For example, Verboven et al use a "polar-zero destructive pair" to represent spurious modes, and Goethals defines four indicators to distinguish between real and spurious modes. Deraemaker et al introduced a modality delivery paradigm to improve the clarity of the stability map, and Lanslots et al verified the feasibility of the CMIF, MPC and MPD indices. Sarlo et al demonstrate how uncertainty becomes a valuable tool in various cases of automation. Cluster analysis, as a second type of method, is widely used as a new method for implementing automated pattern recognition. For example, Magalhaes and the like directly use hierarchical clustering, and Reynders and the like propose a three-stage automatic pattern recognition method combining a plurality of discriminant indexes and K-means fuzzy clustering. Yang et al propose an automatic OMA method based on a feature system implementation algorithm and a two-stage clustering strategy. Mao et al propose an identification method that combines K-means clustering and hierarchical clustering with principal component analysis. Fan et al propose a clustering merging technique for the automatic interpretation of the stability graph. Afsharp et al combine stability criteria and K-means clustering for PSDT-based modal parameter automatic identification. With the development of neural networks and machine learning techniques, Kim et al utilize image recognition techniques to achieve automatic peak extraction. Chang et al propose a swarm intelligence optimization algorithm based on particle swarm optimization for optimizing and identifying modal parameters.
Although there have been many research efforts to achieve automatic modality identification, there is still new development space for automatic modality identification based on PSDT. It still lacks a feature index or automatic clustering method with recognition capability, and the existing automatic recognition method still needs to manually determine a threshold or a clustering range. In addition, the automatic identification method based on the PSDT has not been applied to structural operation modal analysis affected by environmental excitation, and the applicability of the method in a complex engineering environment is difficult to explain.
Disclosure of Invention
The invention mainly aims to overcome the defects in the prior art and provides a mode automatic identification method based on a power spectrum transfer ratio and a support vector machine. The method provided by the invention can automatically operate to identify the modal parameters without manual intervention, can effectively eliminate false modes, can accurately identify the dense mode by the automatic identification method, and has strong robustness to resist noise interference, thereby having wide application prospect in real-time SHM.
The invention adopts the following technical scheme:
a mode automatic identification method based on a power spectrum transfer ratio and a support vector machine is characterized by comprising the following steps:
establishing a transfer ratio function and a transfer ratio matrix;
calculating the gradient of a peak curve of the PSDT rational function;
automatically identifying system poles based on a support vector machine;
solving modal parameters to construct a stable graph;
calculating a modal similarity coefficient;
and automatically identifying a stable axis based on the support vector machine.
Specifically, the establishing of the transfer ratio function and the transfer ratio matrix specifically includes:
for a multiple degree of freedom system, the relationship between input excitation and output response power spectral density in the frequency domain is established as follows:
Syy(s)＝H(s)Guu(s)H*(s)
wherein: s. theyy(s) Power spectral Density matrix, G, representing the structural responseuu(s) is an input to a power spectral density matrix, H(s) represents a frequency response function matrix, H*(s) represents the complex conjugate transpose of H(s).
At the system pole, a convergence solution of the single reference point PSDT function is derived:
Figure GDA0003664057180000041
wherein:
Figure GDA0003664057180000048
represents the output yi(t) and yj(t) output y at the same transfer ratiop(t) transfer ratio, Siq(s) represents the output yi(t) and yp(t) power spectral density matrix, Hik(s) represents the output yi(t) and input uk(k＝1,2,…,Nr) Frequency response function matrix of NrIndicating the number of system inputs, Gkn(s) denotes the effect on degrees of freedom k and N (N ═ 1,2, …, Nr) Power spectral density matrix of phiirAnd phijrRepresenting the mode shape components corresponding to the degrees of freedom i and j.
The subtraction of two PSDT functions with different transmission ratio outputs under the same vibration condition satisfies the following conditions:
Figure GDA0003664057180000042
wherein: lambda [ alpha ]rRepresenting the pole of the system of the r-th order,
Figure GDA0003664057180000043
function representing transfer ratio
Figure GDA0003664057180000044
And
Figure GDA0003664057180000045
the difference of (a).
The pole of the system is the point corresponding to the PSDT difference function zero value; combining PSDT difference functions of different measurement degrees of freedom to obtain PSDT rational function delta T(-1)The following were used:
Figure GDA0003664057180000046
wherein: n is a radical ofoRepresenting the number of system outputs.
Based on the concept of the power spectrum transfer ratio and the characteristics of the power spectrum transfer ratio at the pole point of the system, the PSDT matrix is formed by combining the power spectrum transfer ratios corresponding to different reference points.
Specifically, calculating the slope of the peak curve of the PSDT rational function specifically includes:
the nominal slopes on the left and right sides of the peak are defined as follows:
Figure GDA0003664057180000047
wherein: sLAnd SRNominal slopes, W, to the left and right of the peak, respectivelyLAnd WRNominal width, H, to the left and right of the peak respectivelyLAnd HRThe nominal heights to the left and right of the peak, respectively, are indicated, and N represents the total number of peak points.
The nominal height and the nominal width of the left side and the right side of the peak value are respectively determined by the relative difference value of the peak value point and inflection points on the left side and the right side of the peak value curve, and the inflection points of the peak value curve are determined by the maximum value of a second derivative function of the peak value curve near the peak value.
Specifically, the automatic identification of system poles based on the support vector machine specifically includes:
given a training sample { xi,yi},i＝1,2,…,l，x∈Rd,yiE { -1,1}, there is a hyperplane w · x + b ═ 0 separating the data and satisfying the following equation:
yi[(w·xi)+b]-1≥0
the classification interval is calculated as follows:
Figure GDA0003664057180000051
wherein the maximum interval is 2/| | w | |, and the obtained support vector is used for constructing the optimal hyperplane.
Comparing the maximum value of the gradient on the left side and the right side of the peak value curve with a threshold value, wherein the maximum value of the gradient on the left side and the right side of the peak value curve is as follows:
max{SL(i),SR(i)}＞ε0
i∈{1,2,…,N}
wherein: epsilon0A gradient threshold value representing a peak value curve is determined by the SVM classifier; when the above formula is satisfied, the peak value is taken as a preliminary identification result of the automatic identification method.
Specifically, solving the modal parameter to construct the stable graph specifically includes:
in the frequency domain, the system outputs O, O ═ 1,2, …, NoIn which N isoThe number of points is output, and the system input is described by a right matrix fractional model of a PolyMAX algorithm, and is expressed as:
Ho(ω)＝Uo(ω)Do(ω)-1
wherein:
Figure GDA0003664057180000052
representing all inputs NiAnd a response matrix between the typical outputs (e.g. FRF and PSDT matrices),
Figure GDA0003664057180000053
a row vector representing the molecular polynomial,
Figure GDA0003664057180000054
is a denominator polynomial matrix of Uo(omega) and Do(ω) is represented by the following form:
Figure GDA0003664057180000061
wherein: n is a polynomial order, the polynomial basis function is taken to be:
Figure GDA0003664057180000062
wherein the denominator coefficient matrix
Figure GDA0003664057180000063
Sum molecular coefficient row vector
Figure GDA0003664057180000064
Is the parameter to be estimated.
A right matrix fraction model formula is subjected to linearization processing, and a denominator coefficient matrix is solved by utilizing a least square principle; the dimension of the standard equation is greatly reduced by reducing the standard equation, and the reduced standard equation is as follows:
Figure GDA0003664057180000065
wherein: m is a reduced standard equation defined by the above formula, and alpha is a denominator polynomial
Figure GDA0003664057180000066
Figure GDA0003664057180000067
Figure GDA0003664057180000068
Figure GDA0003664057180000069
Figure GDA00036640571800000610
Wherein:
Figure GDA00036640571800000611
is the product of Kronecker, NfTo identify the number of frequency points contained within a frequency band,
Figure GDA00036640571800000612
the calculated actual frequency response function matrix O line; identifying denominator coefficient matrix Ar。
The characteristic value decomposition is carried out on the unitary matrix to obtain a system pole of
Figure GDA00036640571800000613
Further obtaining the natural frequency omega of the structurerAnd modal damping ratio ζrThe relationship is as follows:
Figure GDA0003664057180000071
wherein: omegar＝[ωr,1,…,ωr,s,…,ωr,N],ζr＝[ζr,1,…,ζr,s,…,ζr,N]；ωr,sAnd ζr,sRepresenting the r-th natural frequency and resistance in the s-th orderAnd (4) the Nibi.
Obtaining modal parameters of each order, calculating system poles of different orders, drawing the system poles on the same graph to form a stable graph of the system poles, wherein the poles representing the real modal stably appear near a certain frequency, and a clear axis is gradually formed and is called as a stable axis.
Specifically, calculating the modal similarity coefficient specifically includes:
in the automatic identification process, the system frequency omega of the ith order mode is identified from the peak curve driven by the PSDTi(ii) a Frequency ω of the s-th mode of the r-th orderr,sPassing through natural frequency omegarCalculating by a calculation formula; the similarity coefficient of the natural frequencies is defined as follows:
Figure GDA0003664057180000072
damping ratio ζ of s-th mode of r-th orderr,sBy modal damping ratio ζrThe similarity coefficient of the damping ratio is defined as follows:
Figure GDA0003664057180000073
specifically, the automatic identification of the stable axis based on the support vector machine specifically includes:
if the similarity coefficient of the natural frequency is smaller than a certain threshold, it is taken as one of the true system poles:
ηf≤ξf,
wherein: xi shapefAnd a similarity coefficient representing the natural frequency is determined by the SVM classifier.
The stability of the damping ratio is judged from the pole meeting the frequency stability requirement, the similarity coefficient of the damping ratio is smaller than a certain threshold value and is classified as a real system pole
ηd≤ξd,
Wherein: xi shapedSimilarity coefficient representing damping ratio, from SVAnd determining an M classifier.
Selecting an SVM classifier to automatically determine a threshold value of the similarity coefficient of the natural frequency and the damping ratio, and selecting the similarity coefficient of the natural frequency and the damping ratio identified from the reference model as training data to generate the SVM classifier; defining the poles corresponding to the similarity coefficients meeting the requirements of the physical mode as physical stable points; otherwise, defining the model as a pseudo pole, and using the trained model for testing the structure and automatically distinguishing the physical stable point and the pseudo pole so as to realize the automatic identification of the stable axis.
As can be seen from the above description of the present invention, compared with the prior art, the present invention has the following advantages:
(1) compared with the traditional modal identification method, the method provided by the invention constructs the structural response matrix based on power spectrum transmission, is insensitive to environmental factor change, has strong robustness, and is more suitable for identifying structural modal parameters in an operating environment.
(2) Compared with the existing automatic modal identification method, the method provided by the invention can automatically operate to identify modal parameters without manual intervention, can effectively eliminate false modes, accurately identifies the dense mode, and has wide application prospect in real-time SHM.
Drawings
FIG. 1 is a schematic diagram of an ASCE-Benchmark model according to an embodiment of the present invention; wherein, figure (a) is a reference model; FIG. (b) beam-column connection; figure (c) an accelerometer;
FIG. 2 is a peak curve of a PSDT matrix drive and an FRF matrix drive according to an embodiment of the present invention;
FIG. 3 is a nominal slope determination process in a PSDT driven peak profile according to an embodiment of the present invention; wherein, plot (a) is single-modal; graph (b) dense mode;
FIG. 4 shows the peak automatic identification result in the peak curve of the PSDT driver according to the embodiment of the present invention;
FIG. 5 shows a stability diagram generated by the PolyMAX method according to an embodiment of the present invention; wherein, graph (a) is a stability graph of FRF drive, and graph (b) is a stability graph of PSDT drive;
FIG. 6 is a stability diagram of a PSDT drive according to an embodiment of the present invention; where plot (a) considers only frequency similarity and plot (b) considers frequency to damping ratio similarity;
FIG. 7 is a layout diagram of an acceleration sensor of a corridor structure according to an embodiment of the invention;
FIG. 8 is a time-course diagram of acceleration collected by the sensor ES-MS-VA1 according to an embodiment of the present invention;
FIG. 9 illustrates a peak plot of the corridor structure driven by the PSDT of the embodiment of the present invention;
FIG. 10 illustrates a diagram of corridor structural stability under PSDT actuation in accordance with an embodiment of the present invention; wherein plot (a) is a plot considering only frequency similarity and plot (b) is a plot considering frequency to damping ratio similarity;
FIG. 11 is a schematic diagram of the first nine frequencies and modes of the corridor configuration according to the embodiment of the present invention;
FIG. 12 is a graph of modal frequency identification results of a corridor structure in an operation phase according to an embodiment of the present invention;
Detailed Description
The invention provides a modal automatic identification method based on a power spectrum transfer ratio and a support vector machine, which comprises the following steps:
s101: establishing a transfer ratio function and a transfer ratio matrix;
the establishing of the transfer ratio function and the transfer ratio matrix specifically includes:
for a multiple degree of freedom system, the relationship between input excitation and output response power spectral density in the frequency domain is established as follows:
Syy(s)＝H(s)Guu(s)H*(s)
wherein: syy(s) Power spectral Density matrix, G, representing the structural responseuu(s) is an input to a power spectral density matrix, H(s) represents a frequency response function matrix, H*(s) represents the complex conjugate transpose of H(s).
At the system pole, a convergence solution of the single reference point PSDT function is derived:
Figure GDA0003664057180000091
wherein:
Figure GDA0003664057180000092
represents the output yi(t) and yj(t) output y at the same transfer ratiop(t) transfer ratio, Siq(s) represents the output yi(t) and yp(t) power spectral density matrix, Hik(s) represents the output yi(t) and input uk(k＝1,2,…,Nr) Frequency response function matrix of NrIndicating the number of system inputs, Gkn(s) denotes the effect on degrees of freedom k and N (N ═ 1,2, …, Nr) Power spectral density matrix of (phi)irAnd phijrRepresenting the mode shape components corresponding to the degrees of freedom i and j.
The subtraction of two PSDT functions with different transmission ratio outputs under the same vibration condition satisfies the following conditions:
Figure GDA0003664057180000101
wherein: lambdarRepresenting the pole of the system of the r-th order,
Figure GDA0003664057180000102
function representing transfer ratio
Figure GDA0003664057180000103
And
Figure GDA0003664057180000104
the difference of (c).
The pole of the system is the point corresponding to the PSDT difference function zero value; combining PSDT difference functions of different measurement degrees of freedom to obtain PSDT rational function delta T(-1)The following were used:
Figure GDA0003664057180000105
wherein: n is a radical ofoRepresenting the number of system outputs.
Based on the concept of the power spectrum transfer ratio and the characteristics of the power spectrum transfer ratio at the pole point of the system, the PSDT matrix is formed by combining the power spectrum transfer ratios corresponding to different reference points.
S102: calculating the gradient of a peak curve of the PSDT rational function;
calculating the gradient of the peak curve of the PSDT rational function, which specifically comprises the following steps:
the nominal slopes on the left and right sides of the peak are defined as follows:
Figure GDA0003664057180000106
wherein: sLAnd SRNominal slopes, W, to the left and right of the peak, respectivelyLAnd WRNominal widths, H, to the left and right of the peak, respectivelyLAnd HRNominal heights to the left and right of the peak, respectively, are indicated, N representing the total number of peak points;
the nominal height and the nominal width of the left side and the right side of the peak value are respectively determined by relative difference values of the peak value point and inflection points on the left side and the right side of the peak value curve, and the inflection points of the peak value curve are determined by the maximum value of a second derivative function of the peak value curve near the peak value.
S103: automatically identifying system poles based on a support vector machine;
the system pole automatic identification based on the support vector machine specifically comprises the following steps:
given training sample { x }i,yi},i＝1,2,…,l，x∈Rd,yiE { -1,1}, there is one hyperplane w · x + b ═ 0 separating the data and satisfying the following equation:
yi[(w·xi)+b]-1≥0
the classification interval is calculated as follows:
Figure GDA0003664057180000111
wherein the maximum interval is 2/| w | |, and the obtained support vector is used for constructing an optimal hyperplane;
comparing the maximum value of the gradient on the left side and the right side of the peak value curve with a threshold value, as follows:
max{SL(i),SR(i)}＞ε0
i∈{1,2,…,N}
wherein: epsilon0A gradient threshold value representing a peak curve, determined by the SVM classifier; when the above formula is satisfied, the peak value is taken as a preliminary identification result of the automatic identification method.
S104: solving modal parameters to construct a stable graph;
solving the modal parameter to construct a stable graph specifically comprises:
in the frequency domain, the system outputs O, O ═ 1,2, …, NoIn which N isoThe number of points is output, and the system input is described by a right matrix fractional model of a PolyMAX algorithm, and the right matrix fractional model is expressed as follows:
Ho(ω)＝Uo(ω)Do(ω)-1
wherein:
Figure GDA0003664057180000112
representing all inputs NiAnd a response matrix between the typical outputs (e.g. FRF and PSDT matrices),
Figure GDA0003664057180000113
a row vector representing the molecular polynomial,
Figure GDA0003664057180000114
is a denominator polynomial matrix, where Uo(omega) and Do(ω) is represented by the following form:
Figure GDA0003664057180000115
wherein: n is the polynomial order, the polynomial basis function is taken as:
Figure GDA0003664057180000121
wherein the denominator coefficient matrix
Figure GDA0003664057180000122
Sum molecular coefficient row vector
Figure GDA0003664057180000123
Is the parameter to be estimated.
Carrying out linear processing on a right matrix fraction model formula, and solving a denominator coefficient matrix by using a least square principle; the dimension of the standard equation is greatly reduced by reducing the standard equation, and the reduced standard equation is as follows:
Figure GDA0003664057180000124
wherein: m is a reduced standard equation defined by the above formula, and alpha is a denominator polynomial
Figure GDA0003664057180000125
Figure GDA0003664057180000126
Figure GDA0003664057180000127
Figure GDA0003664057180000128
Figure GDA0003664057180000129
Wherein:
Figure GDA00036640571800001210
is the product of Kronecker, NfTo identify the number of frequency points contained within a frequency band,
Figure GDA00036640571800001211
the calculated actual frequency response function matrix O line; identifying denominator coefficient matrix Ar；
The characteristic value decomposition is carried out on the unitary matrix to obtain a system pole of
Figure GDA00036640571800001212
Further obtaining the natural frequency omega of the structurerAnd modal damping ratio ζrThe relationship is as follows:
Figure GDA00036640571800001213
wherein: omegar＝[ωr,1,…,ωr,s,…,ωr,N],ζr＝[ζr,1,…,ζr,s,…,ζr,N]；ωr,sAnd ζr,sRepresenting the r-th natural frequency and damping ratio in the s-th order.
Obtaining modal parameters of each order, calculating system poles of different orders, drawing the system poles on the same graph to form a stable graph of the system poles, wherein the poles representing the real mode stably appear near a certain frequency and gradually form a clear axis which is called as a stable axis.
S105: calculating a modal similarity coefficient;
calculating a modal similarity coefficient, specifically comprising:
in the automatic identification process, the system frequency omega of the ith order mode is identified from the peak curve driven by the PSDTi(ii) a Frequency ω of the s-th mode of the r-th orderr,sPassing through the natural frequency omegarCalculating by a calculation formula; the similarity coefficient of the natural frequency is defined as follows:
Figure GDA0003664057180000131
damping ratio ζ of the s-th mode of the r-th orderr,sThrough modal damping ratio ζrThe similarity coefficient of the damping ratio is obtained by the calculation formulaThe definition is as follows:
Figure GDA0003664057180000132
s106: and automatically identifying a stable axis based on the support vector machine.
The method for automatically identifying the stable axis based on the support vector machine specifically comprises the following steps:
if the similarity coefficient of the natural frequency is smaller than a certain threshold, the natural frequency is taken as one of the real system poles:
ηf≤ξf,
wherein: xifThe similarity coefficient, which represents the natural frequency, is determined by the SVM classifier.
The stability of the damping ratio is judged from the pole meeting the frequency stability requirement, the similarity coefficient of the damping ratio is smaller than a certain threshold value and is classified as a real system pole
ηd≤ξd,
Wherein: xidThe similarity coefficient, which represents the damping ratio, is determined by the SVM classifier.
Selecting an SVM classifier to automatically determine a threshold value of the similarity coefficient of the natural frequency and the damping ratio, and selecting the similarity coefficient of the natural frequency and the damping ratio identified from the reference model as training data to generate the SVM classifier; defining the poles corresponding to the similarity coefficients meeting the requirements of the physical mode as physical stable points; otherwise, defining the model as a pseudo pole, and using the trained model for testing the structure and automatically distinguishing the physical stable point and the pseudo pole so as to realize the automatic identification of the stable axis.
Example 1: automatic modal identification of ASCE-Benchmark model
The reference model constructed at university of british columbia is a steel frame with 1/3 reduced sizes, and has 4 layers, 2 × 2 spans, 2.5 × 2.5m plane size, 0.9m layer height, 8 inclined struts on each layer, and the beam columns are fixedly connected, flexibly connected with a support and a structure and can be freely disassembled and assembled, as shown in fig. 1. Nine vertical columns are bolted to a steel foundation frame and the lower flanges of the two foundation beams are wrapped in concrete, thereby fixing the steel frame to the concrete slab. The component is 300W grade hot rolled steel, specially designed for testing structures. 15 accelerometers were placed throughout the frame to measure the response of the test structure. Two accelerometers measure the north-south movement of the structure and one accelerometer measures the east-west movement of the structure. A series of environmental and forced stimuli were performed on the structure, including weight and vibrator tests.
The test environment is an environmental stimulus under non-destructive conditions, and the method is tested with acceleration data collected at a sampling frequency of 200 Hz. The PSDT matrix is calculated and constructed by equation (3), and the PSDT rational function is calculated using equation (4). The peak curve of the PSDT rational function is plotted in fig. 2. For comparison, the peak curves calculated using the FRF function are also plotted. It can be seen that the peaks obtained by the PSDT-driven algorithm are more clearly visualized than those obtained by the FRF-driven algorithm, especially for dense modes in the frequency range of 7-8 Hz.
Possible peaks can be selected using the peak picking method of PSDT driving, but it is still difficult to distinguish which peaks correspond to real or spurious poles. Based on this, the gradient around the peak is calculated as a key index. Using a single mode in the frequency range of 14-15Hz (see fig. 3(a)) as an example, it is explained how to determine the nominal gradient: the peak position at 14.49Hz was determined from the curve of the PSDT rational function, the start position of the peak at 14.21Hz and the end position at 14.73Hz were determined from the second derivative of the PSDT rational function, and then W was determinedL，HLAnd WR，HRTo calculate the nominal slopes of the left and right sides. The same process is performed for dense modes in the frequency range of 7-8Hz, and the result is shown in fig. 3 (b).
The result of the trained SVM classifier applied to the Benchmark model to automatically identify the peak from the PSDT-driven peak curve is shown in fig. 4. The detection area of each possible peak is a position in the vicinity of the peak determined from the start point and the end point. Wherein only the peaks marked as true peaks by the SVM classifier are selected as possible system poles and highlighted with blue dots in the figure. The results show that the method is able to identify dense modes of 7.41Hz and 7.79 Hz.
For comparison, FRF and PSDT matrices are substituted for equation (11-a) to generate the response matrices for FRF and PSDT drives, respectively. The denominator coefficient matrix is determined by reducing the standard equation and the unitary matrix is subjected to eigenvalue decomposition. Finally, the determined characteristic values are substituted for the formula (12) to calculate the corresponding frequency and damping ratio under the system order. The system order is gradually increased from 2 to 100, the above calculation process is repeated, and the frequency and damping ratio corresponding to each system order are obtained. The results of the stability diagram of the FRF drive and the stability diagram of the PSDT drive are shown in fig. 5(a) and 5(b), respectively. These points are highly discrete in the FRF driver stability diagram and it is difficult to determine the stability axis. In contrast, the stabilization points in the PSDT-driven stabilization graph are more concentrated, and a relatively clear stabilization axis can be formed.
Although PSDT-driven stability maps can identify modal information, this nonparametric approach involves many subjective judgments. Based on this, a similarity coefficient of a natural frequency and a damping ratio is defined as an index for distinguishing a true pole and a false pole, and a machine learning technique is introduced to automate it. The similarity coefficients of the natural frequency and the damping ratio obtained from the Benchmark model are used as samples for training the SVM classifier, wherein 1000 samples are used for training, and the other 500 samples are used for testing the accuracy of the model. The model accuracies of the frequency and damping ratio were 97.2% and 98.0%, respectively.
Subsequently, the SVM classifier using the similarity coefficients examines all possible system poles determined in the first stage. First, a similarity coefficient model of the natural frequency is used for calculation, and five stable axes can be generated in a PSDT-driven stable graph, as shown in fig. 6 (a). In addition, the stability pole satisfying the frequency stability requirement is checked again through the similarity coefficient model of the damping ratio. The stability chart results after checking the stability of the damping ratio are shown in fig. 6 (b). The lower order poles of the system are further eliminated because the damping ratio is unstable than would be the result if only frequency stability were considered. Finally, five true modalities may be determined based on the stable axes plotted in fig. 6 (b). Table 1 lists the recognition results using the automatic recognition method.
TABLE 1 Modal parameters identified by ASCE-Benchmark model
Figure GDA0003664057180000161
aW-i, S-i, and T-i are the recognition results of the ith mode in weak, strong direction and twist, respectively.
In order to verify the accuracy of the modal parameters identified by the proposed method, the natural frequencies determined by the different methods are very close to each other, but there is a large difference in the determined damping ratio. In order to further improve the accuracy of the damping estimation, optimization based on a parametric maximum likelihood smoothing method may be considered.
In this validation, acceleration data from the Benchmark model is used as system input, and a well-designed program runs automatically to identify modal parameters without manual intervention. The identification result shows that the method can effectively eliminate the false mode and accurately identify the dense mode.
Example 2: high-rise corridor structure application
The Xiamen national trade finance center is a high-rise building and is positioned in Xiamen of a coastal city in southeast China. The building is provided with two towers with the height of 137m and an overbridge with the length of 42m, and the overbridge is connected with the two towers and forms a structure in the shape of a Chinese character 'men'. Rigid connections are used between the overpass and the tower to increase structural rigidity, but are very sensitive to relative deformations between the towers.
To monitor the security and integrity of complex structures, the xiamen national trade financial center has installed a structural health monitoring system. Its integrated architecture consists of four systems: (1) a sensor system; (2) a data acquisition and transmission system; (3) a data analysis and processing system; (4) a health status assessment system. The sensing system comprises 94 sensors installed in the overpass, including anemometers, temperature sensors, strain sensors, displacement sensors, inclinometers, accelerometers, and hydrostatic levels.
To validate the modal automatic identification method proposed by the present invention, acceleration data collected by an SHMS installed in the corridor structure was used. Six accelerometers, both vertically and horizontally, are mounted in the quarter span, the middle span and the three-quarter span of the corridor, as shown in figure 7.
Taking the acceleration data collected by the accelerometer ES-MS-VA1 as an example, a 300s acceleration time course plot is plotted in FIG. 8, with a sampling frequency of 50 Hz.
In the process of mode identification, the peak value of the peak value curve under the PSDT drive can be automatically identified. The identification result of the PSDT-driven lower peak curve of the high-rise building is shown in fig. 9. The detection areas of all possible peaks are highlighted with a rectangular box. Due to noise interference, many false peaks exist which are difficult to remove by subjective judgment. In order to distinguish real peaks from false peaks, the nominal slope of each peak is examined by a trained SVM classifier. Only peaks meeting the nominal slope requirement are selected as possible system poles and highlighted with a blue dot. At this stage, most spurious modes are eliminated, and some closely spaced modes are retained for further examination.
Possible system poles are further examined to determine if a corresponding stable axis exists. First, a stable point that meets the requirement of the natural frequency similarity coefficient is retained to generate a stable axis, and the SVM classifier of the natural frequency is used for the identification. As shown in fig. 10(a), in the PSDT-driven stabilization diagram of the corridor structure, clear 10 stabilization axes are generated. In addition, a stability pole meeting the frequency stability requirement is checked by a damping ratio SVM classifier. After the stability of the natural frequency and the damping ratio is considered, a stability map of the PSDT drive is regenerated and plotted in fig. 10 (b). Since the stable axis corresponding to the frequency of 0.26Hz is eliminated due to the unstable damping ratio, only 9 stable axes are remained in the stable graph. To verify the authenticity of the identified structural modes, the first nine order frequencies and corresponding mode shapes were determined, as shown in FIG. 11. It can be seen that the recognition results identify well the dense modes of existence of the structure.
To test the robustness of the automatic identification method, 100 sets of acceleration data measured from an accelerometer at 5 minute intervals were analyzed, and the results are shown in fig. 12. The modes are well distinguished, and the modal frequency of each order is relatively stable. The method also proves that the modal automatic identification method provided by the invention has strong robustness and is expected to be applied to real-time health monitoring.
The above description is only an embodiment of the present invention, but the design concept of the present invention is not limited thereto, and any insubstantial modifications made by using the design concept should fall within the scope of infringing the present invention.

Claims (6)

1.A mode automatic identification method based on a power spectrum transfer ratio and a support vector machine is characterized by comprising the following steps:
establishing a transfer ratio function and a transfer ratio matrix;
calculating the gradient of a peak curve of the PSDT rational function;
automatically identifying system poles based on a support vector machine;
solving modal parameters to construct a stable graph;
calculating a modal similarity coefficient;
automatically identifying a stable axis based on a support vector machine;
the automatic identification of the stable axis based on the support vector machine specifically comprises the following steps:
if the similarity coefficient of the natural frequency is smaller than a certain threshold, the natural frequency is taken as one of the real system poles:
ηf≤ξf,
wherein: xi shapefA similarity coefficient representing the natural frequency, determined by an SVM classifier;
the stability of the damping ratio is judged from the pole meeting the frequency stability requirement, the similarity coefficient of the damping ratio is smaller than a certain threshold value and is classified as a real system pole
ηd≤ξd,
Wherein: xidA similarity coefficient representing the damping ratio, determined by an SVM classifier;
selecting an SVM classifier to automatically determine a threshold value of a similarity coefficient of the natural frequency and the damping ratio, and selecting the similarity coefficient of the natural frequency and the damping ratio identified from the reference model as training data to generate the SVM classifier; defining a pole corresponding to the similarity coefficient meeting the requirement of the physical mode as a physical stable point; otherwise, defining the model as a pseudo pole, and using the trained model for testing the structure and automatically distinguishing the physical stable point and the pseudo pole, thereby realizing the automatic identification of the stable axis.
2. The method for automatically identifying the modality based on the power spectrum transfer ratio and the support vector machine according to claim 1, wherein the establishing of the transfer ratio function and the transfer ratio matrix specifically comprises:
for a multiple degree of freedom system, the relationship between input excitation and output response power spectral density in the frequency domain is established as follows:
Syy(s)＝H(s)Guu(s)H*(s)
wherein: syy(s) Power spectral Density matrix, G, representing the structural responseuu(s) is an input to a power spectral density matrix, H(s) represents a frequency response function matrix, H*(s) represents the complex conjugate transpose of H(s);
when s approaches the system pole λ of the order rrAnd deducing a convergence solution of the single reference point PSDT function:
Figure FDA0003649724630000021
wherein:
Figure FDA0003649724630000022
represents the output yi(t) and yj(t) output y at the same transfer ratioq(t) transfer ratio, Siq(s) represents the output yi(t) and yq(t) power spectral density matrix, Hik(s) represents the output yi(t) and input ukK is 1,2, …, Nr，NrIndicating the number of system inputs, Gkn(s) represents the power spectral density matrix acting at the degrees of freedom k and N, N being 1,2, …, Nr，ΦirAnd phijrRepresenting vibration mode components corresponding to the degrees of freedom i and j;
subtracting two PSDT functions with different transmission ratio outputs under the same vibration condition satisfies the following conditions:
Figure FDA0003649724630000023
wherein: lambdarRepresenting the pole of the system of the r-th order,
Figure FDA0003649724630000024
function representing transfer ratio
Figure FDA0003649724630000025
And
Figure FDA0003649724630000026
a difference of (d);
the pole of the system is the point corresponding to the zero value of the PSDT difference function; combining PSDT difference functions of different measurement degrees of freedom to obtain PSDT rational function delta T(-1)The following:
Figure FDA0003649724630000027
wherein: n is a radical of hydrogenoRepresenting the number of system outputs;
based on the concept of the power spectrum transfer ratio and the characteristics of the power spectrum transfer ratio at the pole point of the system, the PSDT matrix is formed by combining the power spectrum transfer ratios corresponding to different reference points.
3. The method according to claim 1, wherein the step of calculating the slope of the peak curve of the PSDT rational function comprises:
the nominal slopes on the left and right sides of the peak are defined as follows:
Figure FDA0003649724630000031
wherein: sLAnd SRNominal slopes, W, to the left and right of the peak, respectivelyLAnd WRNominal widths, H, to the left and right of the peak, respectivelyLAnd HRNominal heights to the left and right of the peak, respectively, are indicated, N representing the total number of peak points;
the nominal height and the nominal width of the left side and the right side of the peak value are respectively determined by relative difference values of the peak value point and inflection points on the left side and the right side of the peak value curve, and the inflection points of the peak value curve are determined by the maximum value of a second derivative function of the peak value curve near the peak value.
4. The method according to claim 3, wherein the automatic modal identification based on the power spectrum transfer ratio and the support vector machine is based on automatic system pole identification of the support vector machine, and specifically comprises:
given a training sample { xi,yi},i＝1,2,…,l，x∈Rd,yiE { -1,1}, there is one hyperplane w · x + b ═ 0 separating the data and satisfying the following equation:
yi[(w·xi)+b]-1≥0
the classification interval is calculated as follows:
Figure FDA0003649724630000032
wherein: the maximum interval is 2/| w | |, and the obtained support vector is used for constructing an optimal hyperplane;
comparing the maximum value of the gradient on the left side and the right side of the peak value curve with a threshold value, as follows:
max{SL(i),SR(i)}＞ε0
i∈{1,2,…,N}
wherein: epsilon0A gradient threshold value representing a peak curve, determined by the SVM classifier; when the above formula is satisfied, the peak value is taken as a preliminary identification result of the automatic identification method.
5. The method according to claim 1, wherein solving modal parameters to construct a stability graph comprises:
in the frequency domain, the system outputs O, O ═ 1,2, …, NoIn which N isoThe number of points is output, and the system input is described by a right matrix fractional model of a PolyMAX algorithm, and is expressed as:
Ho(ω)＝Uo(ω)Do(ω)-1
wherein:
Figure FDA0003649724630000041
representing all inputs NiAnd a response matrix between the typical outputs,
Figure FDA0003649724630000042
a row vector representing the molecular polynomial,
Figure FDA0003649724630000043
is a denominator polynomial matrix, where Uo(omega) and Do(ω) is represented by the form:
Figure FDA0003649724630000044
wherein: n is the polynomial order, the polynomial basis function is taken as:
Figure FDA00036497246300000412
wherein the denominator coefficient matrix
Figure FDA0003649724630000045
Sum molecular coefficient row vector
Figure FDA0003649724630000046
Is a parameter to be estimated;
carrying out linear processing on a right matrix fraction model formula, and solving a denominator coefficient matrix by using a least square principle; the dimension of the standard equation is greatly reduced by reducing the standard equation, and the reduced standard equation is as follows:
Figure FDA0003649724630000047
wherein: m is a reduced standard equation defined by the above formula, and alpha is a denominator polynomial;
Figure FDA0003649724630000048
Figure FDA0003649724630000049
Figure FDA00036497246300000410
Figure FDA00036497246300000411
Figure FDA0003649724630000051
wherein:
Figure FDA0003649724630000052
is the product of Kronecker, NfTo identify the number of frequency points contained within a frequency band,
Figure FDA0003649724630000053
calculating the obtained actual frequency response function matrix O row; a. therIs a denominator coefficient matrix;
the unitary matrix is subjected to eigenvalue decomposition to obtain a system pole of
Figure FDA0003649724630000054
And obtain the natural frequency omega of the structurerAnd modal damping ratio ζrThe relationship is as follows:
Figure FDA0003649724630000055
wherein: omegar＝[ωr,1,…,ωr,s,…,ωr,N],ζr＝[ζr,1,…,ζr,s,…,ζr,N]；ωr,sAnd ζr,sRepresenting the r natural frequency and damping ratio in the s-th order;
obtaining modal parameters of each order, calculating system poles of different orders, drawing the system poles on the same graph to form a stable graph of the system poles, wherein the poles representing the real mode stably appear near a certain frequency and gradually form a clear axis which is called as a stable axis.
6. The method according to claim 5, wherein the calculating of the modal similarity coefficient specifically includes:
in the automatic identification process, the system frequency omega of the ith order mode is identified from the peak curve driven by the PSDTi(ii) a Frequency ω of the s-th mode of the r-th orderr,sPassing through natural frequency omegarCalculating by a calculation formula; the similarity coefficient of the natural frequencies is defined as follows:
Figure FDA0003649724630000056
damping ratio ζ of the s-th mode of the r-th orderr,sBy modal damping ratio ζrThe similarity coefficient of the damping ratio is defined as follows:
Figure FDA0003649724630000057
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